Mathematics 506 SCI

Mid-Year Examination

January/February 2012

Administration & Marking Guide

565-506

Administration Guide

Design Team: EMSB

1. Presentation of the evaluation situation

This evaluation situation is consistent with principles regarding the evaluation of learning as outlined by the Ministry of Education, Leisure and Sport (MELS). Developed in conjunction with teachers as well as consultants from various school boards in Quebec, it is intended to give teachers some indication of the extent to which students have developed the competency Uses Mathematical Reasoning. This evaluation situation comprises three parts. Part A consists of six multiple choice questions and Part B consists of four short answer questions. These sections are intended to evaluate the student’s mastery of mathematical concepts and processes. Part C consists of six application tasks that focus on evaluating the competency Uses Mathematical Reasoning. The preliminary result for Part C is expressed as a mark out of 600, whereas the final result is expressed as a mark out of 60 which is calculated by dividing the preliminary result by 10 and rounding to the nearest unit. The student's total mark is the sum of results for Parts A, B, and C.

Type of Task Number of Questions Marks per Question Total Marks

Part A Multiple Choice 6 4 24

Part B Short Answer 4 4 16

Part C Application 6 10 60

Tasks in this evaluation situation focus on the main concepts and processes covered in year three of the Secondary Cycle Two Mathematics Program: Science Option. This guide provides information about scoring student work on tasks that make up the evaluation situation. This guide also includes examples of appropriate solutions, as well as Observable Elements associated with appropriate reasoning, and the Maximum level for different examples of student work for each task in Part C. Appendices include a rubric for the competency Uses Mathematical Reasoning (Appendix A), and a chart that provides an overview of student results (Appendix B).

1.1 Description of the documents

The following documents are provided as part of this evaluation situation:

One (1) Administration and Marking Guide which contains a description of the administration conditions as well as the marking key for the student tasks.

One (1) Student Booklet for the situations focusing on Competencies 2 (Uses Mathematical Reasoning).

1.2 Description of the tasks: connections to the Quebec Education Program (QEP)

Description of Part C For each task, the table below gives a brief description of the concepts and processes that a student may be required to mobilize in order to demonstrate the level of their competency Uses Mathematical Reasoning.

Title of the situation Concepts and processes

Question 11

The energy of a spring

Square root function

System of equations

Determines the value or data by solving equations and inequalities

Question 12

The temperature variation of a liquid

Absolute value function

Solving absolute value equations in one variable

Interpreting and representing real-world situations using different registers of representation

Question 13

The profit ratio of a company

Rational function

Linear function

Solving rational equations in one variable

Interpreting and representing real-world situations using different registers of representation

Question 14

The School Dance

Optimizing a situation, taking into account different constraints

Solving a system of inequalities: algebraically or graphically

Choosing one or more optimal solutions Analyzing and interpreting the solution(s), depending on the context

Question 15

The research on bacteria

Exponential function

Solving exponential equations in one variable using logarithms

Question 16

Properties of Logarithms

Manipulating algebraic expressions, using the

properties of logarithms

2. Timetable for administering the examination and time allotted for the

evaluation situations This evaluation situation should be administered in one 3 hour time block on or after February 3rd 2012.

3. Possible adaptations1

Adaptations are permitted for this evaluation situation, but none of them involve changing its content. In fact, any change in the content of the evaluation situation, such as removing or changing a requirement, would compromise its validity.2 Adaptations are intended for students with learning difficulties or social maladjustments, or for students who have temporary limitations due to illness or special circumstances. The school could make these adaptations for students who require special measures. It should be noted that adaptations must always be made in order to allow a student to demonstrate their level of competency development, but must in no way compromise the validity of the evaluation situation. In other words, the adaptations should consist of measures related to the administration of the evaluation situation, its format, or the way in which students submit their work.

4. Procedure for administering the evaluation situation

4.1 Initial preparation

Ask the students to draw up a memory aid. Students may use a memory aid that they have prepared for another evaluation situation if it is the original hand-written copy.

Review the evaluation criteria with the students and explain the indicators for each criterion. For this purpose, you may copy the evaluation grids (Appendix A) onto transparencies.

Remind them that any required calculations or explanations will be taken into account in grading their work in part C.

Remind students that in Part C the scorer must give a mark of 0 to students who fail to show their work or whose work does not justify their answer.

4.2 General procedure

Materials for each student • Student Booklet • Calculator (with or without a graphic display) • Geometry set (ruler, compass, protractor, etc.) • Memory aid

1 The information in this section is based on work in progress being carried out by Doris Tremblay for the MELS.

2 An instrument’s ability to measure what it is designed to measure. (Renald Legendre, Dictionnaire actuel de

l’éducation, Montréal, Guérin, 2005, p. 1436.

5. Administration of the evaluation situation:

• Hand out copies of the Question/Answer Booklet. Ask students to go through the document, familiarizing themselves with all of the information and requirements. Make sure they know where they must write their names, answers, calculations, and explanations.

• Ask students to read the evaluation criteria used to evaluate their level of competency

development. • Describe the basic rules for evaluation situations:

o Each student works alone. o Students may use a calculator but are expected to indicate the sequence of

operations involved as part of the justification for their solution. o Students have three hours to complete the evaluation situation. o Students may use resources such as a geometry set, graph paper and the

handwritten memory aid. o During the evaluation situation, the teacher may clarify the meaning of general

vocabulary related to the context of the task. • When time is up, collect the examination booklets.

6. Using the results

The procedure outlined below should be used to help teachers evaluate the student's mathematical competency Uses Mathematical Reasoning by taking into account the information collected during the administration of the evaluation situation. Examples of appropriate reasoning are given for each task. The student’s reasoning may be different, yet still meet the requirements of the task. The scorer must exercise judgment and accept other appropriate reasoning. A judgment regarding the student’s work on each task is made by taking into account the student’s performance level for each evaluation criterion considered. The level achieved for criterion 3 is the maximum possible level for criteria 2, 4 and 5. The scorer must give a mark of zero to students who fail to show their work or whose work does not justify their answer.

The preliminary result for Part C is expressed as a mark out of 600, whereas the final result is expressed as a mark out of 60 which is calculated by dividing the preliminary result by 10 and rounding to the nearest unit. The student's total mark is the sum of results for Parts A, B, and C. Referring to this evaluation situation, make a judgment regarding the student's level of competency development for the competency Uses Mathematical Reasoning.

7. Marking Key

PART A: Multiple-Choice Questions Questions 1 to 6 4 marks or 0 marks

1. A 4. B 2. B 5. C 3. D 6. B

PART B: Short-Constructed Answer Questions

Questions 7 to 10

7. The equation of the horizontal asymptote is y = 2.

The equation of the vertical asymptote is x = 3.

2 marks each correct answer 0 mark each incorrect answer

8. a)

b) The rule for the composition of g (f(x)) is

2 marks each correct answer 0 mark each incorrect answer

9. The system of inequalities is:

; ; ;

1 mark for each correct answer

10. The solution set of the equation is S=

4 marks correct answer 2 marks 1 correct value 0 mark incorrect answer

PART C: Extended Application Questions Questions 11 to 16 10 marks each (marked on 100% each according to rubric)

11. THE ENERGY OF A SPRING

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF PARAMETERS OF EQUATION

Using (e, S(e)) = (0.04, 0.12),

Using (e, S(e)) = (0.0, 0.13),

Therefore,

So,

DETERMINATION OF THE ELASTIC POTENTIAL ENERGY OF THE SPRING

Using S(e) = 0.165,

CONCLUSION

The elastic potential energy of the spring when the stretched length is 0.165 m is 0.4225 J.

12. THE TEMPERATURE VARIATION OF A LIQUID

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF THE PARAMETERS OF THE FUNCTION

Parameter h: Using the symmetry of the function and the points (6, 12) and (54, 12),

Parameter a: Determining the slope of the left branch of the function by using the points (0, 20) and (6,12),

Because the function opens up,

Parameter k: Using the equation of the left branch of the function and the value of h,

Equation of the left branch:

Solving for x:

Therefore,

So, the equation of the absolute value function is

DETERMINATION OF THE TIME SPENT BELOW 0OC

Therefore, the time spent below 0oC is

So, t =30 minutes

CONCLUSION

The time the liquid spent below 0oC is 30 minutes.

13. THE PROFIT RATIO OF A COMPANY

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF THE EQUATIONS FOR THE NUMERATOR AND DENOMINATOR

Note: Year 2004 will be used as year 0. Numerator, Function E(t):

Therefore, Denominator, Function C(t):

Therefore,

EQUATION OF THE PROFIT RATIO FUNCTION (RATIONAL FUNCTION)

Function P(t)

DETERMINATION OF THE YEAR WHEN THE COMPANY WILL REACH ITS OBJECTIVE

The objective is

So, the company will reach its target objective in 12.5 years since 2004.

CONCLUSION

The company will reach its target objective in 12.5 years since 2004 (during 2017).

14. THE SCHOOL DANCE

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF THE MAXIMUM PROFIT LAST YEAR

Function rule: P = 7x + 10y

POLYGON OF CONSTRAINTS OF ATTENDANCE THIS YEAR

New constraint : x + y ≤ 200. There are two new vertices in the polygon of constraints.

Equation of QR: Slope = (120-80)/(160-80) = ½ Initial value: 80 = ½(80) + b; b = 40 Point A: x + y = 200 and x - 2y= -80; 3y = 280 y = 280/3 x = 320/3

Point A (106, 93) (allow any possible rounding) Equation of SR : Slope = (120-20)/(160-120) = 5/2; Initial value : 20 = 5/2(120) + b; b = -280 Point B: x + y = 200 and 5x – 2y = 560; 7x = 960 x = 960/7 y = 440/7

Point B (137, 62) (allow any possible rounding)

VERTEX REVENUE : 7 x + 10 y

P(40,40) $680

Q(80,80) $1 360

R(160,120) $2 320 Maximum

S(120,20) $1 040

DETERMINATION OF THE MAXIMUM PROFIT THIS YEAR

Function rule: P = 7x + 10y

DETERMINATION OF THE DIFFERENCE IN PROFIT THIS YEAR

Difference = $1 672 - $2 320 = -$648

CONCLUSION

The difference in profit this year is - $648.

VERTEX REVENUE : 7 x + 10 y

P(40,40) $680

Q(80,80) $1 360

A(106,93) $1 672 Maximum

B(137,62) $1 579

S(120,20) $1 040

15. THE RESEARCH ON BACTERIA

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF THE EQUATIONS FOR THE TWO POPULATIONS

Let x be the time in hours

First population:

Parameters: a=500; c=2; b=2

Second population:

Parameters: a=4000; c=0.25; b=3

or

DETERMINATION OF TIME WHEN BOTH POPULATIONS WILL BE EQUAL

(

(

Time: 0.375 h or (0.375h x 60 min/h = 22.5 min)

CONCLUSION

It will take 0.375 hours or 22.5 minutes for both populations to be equal.

16. PROPERTIES OF LOGARITHMS

EXAMPLE OF AN APPROPRIATE SOLUTION

PROVING THAT THE TWO EXPRESSIONS ARE EQUIVALENT

To show that the two expressions are equivalent, you must transform one to obtain the other.

=

= = = = =

Appendix

Evalu

ati

on

Cri

teri

a

Descriptive Chart for Evaluating Competency Appendix A Uses Mathematical Reasoning

Observable Indicators of Student Behaviour

Level 5 Level 4 Level 3 Level 2 Level 1

Cr3

Proper application of

mathematical reasoning

suited to the situation

Takes every aspect of the situation into account.

Uses efficient strategies in applying his/her mathematical reasoning.

Uses mathematical concepts and processes that enable him/her to meet the requirements of the situation efficiently.

Takes the main aspects of the situation into account.

Uses effective strategies in applying his/her mathematical reasoning.

Uses mathematical concepts and processes appropriate for the situation.

Takes some aspects of the situation into account.

Uses a few effective strategies for certain steps in applying his/her mathematical reasoning.

Uses some mathematical concepts and processes appropriate for the situation.

Takes few aspects of the situation into account.

Uses few appropriate strategies in applying his/her mathematical reasoning.

Uses very few mathematical concepts and processes appropriate for the situation.

Takes no aspect of the situation into account.

Uses inappropriate strategies in applying his/her mathematical reasoning.

Uses mathematical concepts and processes that are inappropriate for the situation.

Cr1

Formulation of a conjecture

appropriate to the situation

Formulates an astute conjecture based on a rigorous analysis of the situation or on examples that consider every aspect of a situation.

Formulates an appropriate conjecture based on a fitting analysis of the situation or on examples that consider most of the important aspects of the situation.

Formulates a partially appropriate conjecture based on an analysis of the situation or on examples that consider some aspects of the situation.

Formulates a conjecture that is not very appropriate, based on an analysis that considers few aspects of the situation, or on examples chosen purely by chance.

Formulates a conjecture that is unrelated to the situation.

Cr2

Correct use of concepts and

processes appropriate to the

situation

Applies the chosen mathematical concepts and processes appropriately.

Applies the chosen mathematical concepts and processes appropriately, but makes minor errors (e.g. miscalculations, inaccuracies, omissions).

Applies the chosen mathematical concepts and processes, but makes some conceptual or procedural errors.

Applies the chosen mathematical concepts and processes, but makes several conceptual or procedural errors.

Applies mathematical concepts and processes inappropriately, making many conceptual or procedural errors.

Cr4

Proper organization of the

steps in an appropriate

procedure

Presents a complete and organized procedure that explicitly outlines what was done or how it was done.

Presents a complete and organized procedure that explicitly outlines what was done or how it was done, even though some of the steps are implicit.

Presents a procedure that is not very explicit about what was done or how it was done, because the work is unclear or not very organized.

Presents a procedure consisting of isolated elements, showing little or no work that explicitly outlines what was done or how it was done.

Presents a procedure that is completely unrelated to the situation or does not show any procedure.

Cr5

Correct justification of the

steps in an appropriate

procedure

When required to justify or support his/her statements, conclusions or results, uses solid mathematical arguments.

Rigorously observes the rules and conventions of mathematical language.

When required to justify or support his/her statements, conclusions or results, uses appropriate mathematical arguments.

Observes the rules and conventions of mathematical language, despite some minor errors or omissions.

When required to justify or support his/her statements, conclusions or results, uses some appropriate mathematical arguments or uses rudimentary mathematical arguments.

Makes some errors or is sometimes inaccurate in using the rules and conventions of mathematical language.

When required to justify or support his/her statements, conclusions or results, uses only slightly appropriate mathematical arguments.

Makes several errors related to the rules and conventions of mathematical language.

When required to justify or support his/her statements, conclusions or results, uses erroneous or inappropriate mathematical arguments

Shows little or no concern for the rules and conventions of mathematical language.